Discrete Triebel-Lizorkin spaces and expansive matrices

We provide a characterization of two expansive dilation matrices yielding equal discrete anisotropic Triebel-Lizorkin spaces. For two such matrices A and B, and arbitrary α ∈ R and p, q ∈ (0,∞], it is shown that ˙fαp,q(A) = ˙fα p,q(B) if and only if the set {AjB−j: j ∈ Z} is finite, or in the trivial case when | det(A)|α+1/2−1/p = | det(B)|α+1/2−1/p and p = q. This provides an extension of a result by Triebel for diagonal dilations to arbitrary expansive matrices. The obtained classification of dilations is different from corresponding results for anisotropic Triebel-Lizorkin function spaces.